How can I find the critical points and classify them for multivariable functions using partial derivatives and the second derivative test?

To find and classify critical points of a multivariable function, first compute the partial derivatives and set them to zero to find critical points. Use the second derivative test by evaluating the Hessian matrix at these points. If the Hessian is positive definite, the point is a local minimum; if negative definite, a local maximum; if indefinite, a saddle point.

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